On spectral decomposition of Smale-Vietoris axiom A diffeomorphisms

TL;DR

This paper studies the spectral decomposition of Smale-Vietoris axiom A diffeomorphisms, establishing their relation to classical DE-mappings and analyzing bifurcations of solenoidal basic sets in 3-manifolds.

Contribution

It introduces Smale-Vietoris diffeomorphisms, connects their basic sets to those of axiom A endomorphisms, and proves the uniqueness and bifurcation phenomena of solenoidal basic sets.

Findings

Established correspondence between basic sets of Smale-Vietoris diffeomorphisms and axiom A endomorphisms

Proved uniqueness of nontrivial solenoidal basic set in 3-manifolds

Constructed bifurcation scenarios for solenoidal basic sets

Abstract

We introduce Smale-Vietoris diffeomorphisms that include the classical DE-mappings with Smale solenoids. We describe the correspondence between basic sets of axiom A Smale-Vietoris diffeomorphisms and basic sets of nonsingular axiom A endomorphisms. For Smale-Vietoris diffeomorphisms of 3-manifolds, we prove the uniqueness of nontrivial solenoidal basic set. We construct a bifurcation between different types of solenoidal basic sets which can be considered as a destruction (or birth) of Smale solenoid.